Problem 19 - College Algebra

Question

For each of the following exercises, find the distance between the two points. Simplify your answers, and write the exact answer in simplest radical form for irrational answers. $$(5,0)\text { and }(5,6)$$

Step-by-Step Solution

Answer

The distance between the points is 6.

Step 1: Identify the Formula for Distance

To find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Step 2: Substitute the Coordinates

Substitute the given coordinates \((5, 0)\) and \((5, 6)\) into the distance formula: \(d = \sqrt{(5 - 5)^2 + (6 - 0)^2}\).

Step 3: Calculate the Differences

Calculate \((5 - 5)^2\) which is 0, and \((6 - 0)^2\) which is 36.

Step 4: Simplify the Expression

The expression inside the square root becomes: \(0 + 36\).

Step 5: Calculate the Square Root

Simplify \(\sqrt{36}\) which gives 6.

Key Concepts

Coordinate Geometry

Coordinate geometry combines algebra and geometry to study the position and properties of shapes on a plane. This approach is called analytic geometry. By using a coordinate grid, like the Cartesian coordinate system, you can pinpoint locations, calculate distances, and analyze geometric relationships.

Coordinates are expressed as ordered pairs: \(x, y\). The x-coordinate indicates the horizontal position, while the y-coordinate indicates the vertical position. For example, point (5, 0) is 5 units to the right of the origin along the x-axis.

Understanding and mastering coordinate geometry allows us to solve problems involving distances, angles, and more using algebraic methods, making it an essential tool in mathematics.

Simplifying Radicals

Simplifying radicals is an important step to expressing an answer in its simplest form. A radical expression generally involves a square root, but it can also include cube roots and beyond.

With square roots, the goal is to express the number inside the radical as a product of perfect squares. For example, for the expression \(\sqrt{36}\), 36 is a perfect square because 6 \(\times\) 6 equals 36.

Thus, \(\sqrt{36} = 6\).

When simplifying radicals, remember:

• Look for numbers that can be factored into perfect squares.
• Rewrite the square root as a product of known values.
• Simplify to remove the radical if possible.

This process helps present a cleaner, more understandable answer in mathematics.

Calculation Steps

Calculating the distance between two points involves several straightforward steps. Let's break them down:

First, identify the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Next, substitute the given coordinates. For points (5,0) and (5,6), the formula becomes:

• Calculate \(x_2 - x_1\) which is \(5 - 5 = 0\).
• Calculate \(y_2 - y_1\) which is \(6 - 0 = 6\).

This results in the expression: \(\sqrt{0^2 + 6^2}\), also written as \(\sqrt{0 + 36}\).

Finally, simplify the expression to \(\sqrt{36} = 6\).

Following these steps systematically ensures you always arrive at the correct solution efficiently.